We now introduce a special orthonormal basis in \(L_{2}(-d,d),\)  which was
first discovered in [1]}, also, see [2]}.
Consider a linearly independent set of functions \(\lbrace \alpha ^{n}e^{\alpha }\rbrace _{n=0}^{\infty }\subset L_{2}(-d,d)\) , which is complete in \(L_{2}(-d,d)\) .
The Gram-Schmidt orthonormalization procedure being applied to this set,
results in the orthonormal basis \(\lbrace \Psi _{n}\left( \alpha \right)\rbrace _{n=0}^{\infty }\)  in \(L_{2}(-d,d)\) . The Gram-Schmidt procedure is unstable
when it is applied to an infinite number of functions. However, we have not
seen an instability when applying it to a relatively small number of
functions for \(n\in \left[ 0,12\right] .\)  The same was observed in a number
of other works of this research group, see, e.g. [3]}, [4]}, [5]},
[2]}.
